# Verify Gauss divergence theorem of F=xi+yj+zk over the sphere x^2+y^2+z^2=a^2

Verify Gauss divergence theorem of $F=xi+yj+zk$ over the sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$
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Ashlynn Delacruz
Let ${S}_{a}$ be your sphere, and ${B}_{a}$ the enclosed ball. For each point $\mathbf{r}\in {S}_{a}$ the outwards unit normal n is given by $\mathbf{n}=\frac{\mathbf{r}}{a}$. Furthermore $\mathbf{F}\left(\mathbf{r}\right)=\mathbf{r}$. Since $\mathbf{r}\cdot \mathbf{r}={a}^{2}$ on ${S}_{a}$ it follows that

On the other hand, $\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathbf{F}\right)\equiv 3$, and therefore